GriD-LMIA: A Gridding-Based Assembler for Solving Differentiable Parameter-Dependent Linear Matrix Inequalities

2026-08-04

AI Summary

This study addresses the challenge of time-varying parameter-dependent linear matrix inequalities (PD-LMIs), where derivatives of decision variables induce infinitely many constraints that are difficult to handle. The authors propose a novel approach that discretizes the parameter domain using a hyperrectangular grid and uniformly represents both known functions and decision variables over each grid cell via tensor-product Bernstein polynomials. This transformation converts the infinite constraint set into a finite collection of sufficient LMI conditions. The method innovatively integrates direct Bernstein expansion, Pólya’s theorem-based lifting, and sum-of-squares certificates for the first time, enabling flexible control—within the YALMIP framework—over grid density, polynomial degree, and certificate type. Numerical experiments demonstrate that the proposed technique efficiently solves differentiable PD-LMI problems while achieving a favorable trade-off between computational complexity and solution accuracy.

Abstract

Parameter-dependent linear matrix inequalities (PD-LMIs) require holding over a continuous domain. When the scheduling parameters vary with time, derivatives of parameter-dependent decisions may also enter the conditions. Since semidefinite programming solvers require finitely many constraints, we introduce GriD-LMIA, the Gridding-based Differentiable PD-LMI Assembler. It converts the conditions that need to hold on a continuous domain into finitely many sufficient LMIs in MATLAB. It first partitions the domain with a hyper-rectangular grid and represents known data and decisions on each cell with tensor Bernstein polynomials. The package exports direct Bernstein, Pólya-elevated, and sum-of-squares-based certificates through YALMIP. Examples are given to examine the balance among grid density, decision degree, and certificate choice.

Problem

• parameter-dependent linear matrix inequalities
• continuous domain
• differentiable constraints
• semidefinite programming
• gridding

Innovation

• Gridding-based
• Parameter-dependent LMIs
• Tensor Bernstein polynomials
• Differentiable PD-LMIs
• SOS certificates